NONSMOOTH DATA ERROR ESTIMATES WITH

APPLICATIONS TO THE STUDY OF THE LONG-TIME

BEHAVIOR OF FINITE ELEMENT SOLUTIONS

OF SEMILINEAR PARABOLIC PROBLEMS

Stig Larsson

Abstract. A rather general semilinear parabolic problem is studied together with its spatially semidiscrete finite element approximation. Both problems are formulated within the framework of nonlinear semigroups in the Sobolev space H^{1}(Ω). The main result is an error estimate for solutions with initial data in H^{1}(Ω), valid during an arbitrary finite time interval. The proof is based on the semigroup formulation and is rather elementary. A completely discrete scheme based on the backward Euler method for time discretization is analyzed in an analogous way. This type of error estimate is a key ingredient in recent work by several authors concerning the long-time behavior of finite element solutions. Two examples of this kind are presented: the upper semicontinuity of attractors and a long-time error bound for exponentially stable solutions.

1. Introduction. Recent work by several authors is concerned with the long-time behavior of numerical approximations of semilinear parabolic problems: long-time error bounds [12], [13], convergence of attractors [8], [9], [15], [5], behavior near stationary points [1], [16]. A common feature of these works is the need for an error bound for solutions with nonsmooth initial data. In order to be specific let us describe the idea in the context of a spatially semidiscrete finite element approximation. The continuous problem defines a nonlinear semigroup S(t) in a subspace V of the Sobolev space H^{1}(Ω). The discrete problem defines a nonlinear semigroup S_{h}(t) in a finite dimensional subspace V_{h} of V, h being a discretization parameter. A key ingredient of the works mentioned above is an estimate of the distance between two trajectories S(t)v and S_{h}(t)v with the same initial value v ∈ V_{h}. More precisely one needs an error bound of the following form: Let B be a bounded subset of H^{1}(Ω) and T > 0. For any v ∈ B ∩ V_{h} we have

∥S_{h}(t)v − S(t)v∥_{H^{1}} ≤ C(B, T)h^{p}t^{−σ}, t ∈ (0, T],

for some p, σ > 0. The singularity t^{−σ} arises because the initial regularity is insufficient for the error bound to hold uniformly as t → 0.

In the works mentioned above analogs of this error bound are either postulated or proved, sometimes with sub-optimal order of convergence. The purpose of the present paper is to prove such error bounds in the context of the standard piecewise linear finite element method. We consider both the spatially semidiscrete case and a fully discrete scheme based on the backward Euler method. We obtain results which are optimal both with respect to order of convergence and to regularity requirements. The

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 65M15, 65M60.

Key words and phrases. Semilinear parabolic problem, finite element method, backward Euler method, error estimate, nonsmooth initial data, long-time behavior, analytic semigroup, nonlinear semigroup, upper semicontinuity of attractors.

proofs are based entirely on the semigroup formulation and are rather elementary. For completeness we also include error estimates for solutions with smooth initial data.

Error bounds for finite element approximations of solutions of semilinear parabolic problems with nonsmooth initial data have been given before in, for example, [14], [4], [3], [10], but not in the specific form required here. Also, by restricting our attention to piecewise linear finite elements, and to initial data in H^{1}(Ω) rather than L_{2}(Ω), we are able to give simpler proofs. In [14], [4], [3] it is assumed that the nonlinearity is globally Lipschitz, but here we only require that it satisfies a local Lipschitz condition with a polynomial bound on the constant.

In Section 2 we present the continuous and the spatially semidiscrete problems, and put them into a common abstract framework. In Section 3 we introduce our technique for error estimation in the semidiscrete case, where everything can be presented in a simple way. In Section 4 we review briefly two applications of the error bound from Section 3: a long-time error estimate for exponentially stable solutions (following [12,13]), and upper semicontinuity of attractors (following [8]). Finally in Section 5 we show that the technique from Section 3 can be used also for a completely discrete scheme.

We assume that the reader is familiar with the theory of analytic semigroups, [11, Chapter 1] or [17, Section 2.5], and the corresponding treatment of semilinear parabolic problems, [11, Chapter 3] or [17, Section 6.3].

2. Semigroup formulation. We consider the following semilinear parabolic problem:

(2.1)  \begin{aligned}u_{t} + Au & = f(x, u), & x ∈ Ω, t > 0, \\ u & = 0, & x ∈ ∂Ω, t > 0, \\ u(x, 0) & = u_{0}, & x ∈ Ω, \end{aligned}

where Ω is a bounded domain in R^{d}, d = 1, 2 or 3, u = u(x, t) and u_{t} = ∂u/∂t. We assume that ∂Ω is either smooth or a convex polygon. The operator A is defined by

Au = − ∑_{i,j=1}^{d} \frac{∂}{∂x_{i}} (a_{ij}(x)\frac{∂u}{∂x_{j}} ) + ∑_{i=1}^{d} a_{i}(x)\frac{∂u}{∂x_{i}} + a_{0}(x)u,

where the coefficients a_{ij}, a_{i} are assumed to be continuously differentiable in \overline{Ω} with the matrix a_{ij} symmetric and uniformly positive definite, i.e.,

(2.2)  ∑_{i,j=1}^{d} a_{ij}(x)ξ_{i}ξ_{j} ≥ c_{1}|ξ|^{2},  ∀ξ ∈ R^{d}, x ∈ \overline{Ω}, with c_{1} > 0.

We assume that f is continuous and satisfies a local Lipschitz condition with respect to u. If d = 2 or 3 we assume in addition that the Lipschitz condition is of the form

(2.3)  |f(x, u) − f(x, v)| ≤ C (1 + |u|^{γ} + |v|^{γ})|u − v|,  ∀u, v ∈ R, x ∈ \overline{Ω},

with γ = 2 if d = 3, and γ ∈ [0, ∞) if d = 2.

In the sequel we let (·, ·) and ∥ · ∥ denote the usual inner product and norm in the Hilbert space L_{2}(Ω). The norms in the Sobolev spaces H^{s}(Ω), s ≥ 0, are denoted by ∥ · ∥_{s}. V = H_{0}^{1}(Ω) is the subspace of H^{1}(Ω) consisting of the functions that vanish on ∂Ω, equipped with the norm ∥ · ∥_{1}. V′ = H^{−1}(Ω) is the dual space of V with norm ∥v∥_{−1} = sup_{χ∈V} |(v, χ)|/∥χ∥_{1}.

With the differential operator A we associate the bilinear form

a(u, v) = ∫_{Ω}( ∑_{i,j=1}^{d} a_{ij} \frac{∂u}{∂x_{j}} \frac{∂\overline{v}}{∂x_{i}} + ∑_{i=1}^{d} a_{i} \frac{∂u}{∂x_{i}} \overline{v} + a_{0}u\overline{v}) dx, u, v ∈ V.

By Gårding’s inequality there are constants c_{0} ∈ R and c_{1} > 0 such that

Re a(v, v) + c_{0}∥v∥^{2} ≥ c_{1}∥v∥_{1}^{2}, ∀v ∈ V.

In fact, c_{1} is as in (2.2) and c_{0} = sup_{x∈Ω}(\frac{1}{2} ∑_{j=1}^{d} ∂a_{j}/∂x_{j} − a_{0}), see for example [19, p. 50]. By adding the term c_{0}u to both members of the differential equation in (2.1), we see that we may assume that

(2.4)  Re a(v, v) ≥ c_{1}∥v∥_{1}^{2},  ∀v ∈ V, with c_{1} > 0.

By Green’s formula we have (Au, v) = a(u, v) for u, v ∈ V ∩ H^{2}(Ω), and the weak form of problem (2.1) is to find u(t) ∈ V such that

\begin{matrix}(u_{t}, χ) + a(u, χ) = (f(u), χ), ∀χ ∈ V, t > 0, \\ u(0) = u_{0}, \end{matrix}

where now, with a slight abuse of notation, f denotes the nonlinear operator u ↦ f(·, u(·)).

In view of the elliptic regularity estimate

(2.5)  ∥A^{−1}f∥_{2} ≤ C∥f∥,  f ∈ L_{2}(Ω),

which holds for smooth or convex polygonal domains, we may also consider the operator A as an unbounded operator on L_{2}(Ω) with D(A) = V ∩ H^{2}(Ω). Problem (2.1) then becomes

(2.6)  u_{t} + Au = f(u), t > 0;  u(0) = u_{0}.

We now turn to the discretization of (2.1) with respect to the spatial variables by means of the standard piecewise linear finite element method. We thus let {V_{h}}_{h>0} be a family of finite dimensional subspaces of V, where each V_{h} consists of the continuous piecewise polynomials of degree ≤ 1 with respect to a “triangulation” of the domain Ω with maximum mesh size h. The semidiscrete solution u_{h}(t) ∈ V_{h} satisfies

(2.7)  \begin{matrix}(u_{h,t}, χ) + a(u_{h}, χ) = (f(u_{h}), χ),  ∀χ ∈ V_{h}, t > 0, \\ u_{h}(0) = u_{0h}. \end{matrix}

Introducing the linear operator A_{h} : V_{h} → V_{h} defined by the relation

(A_{h}ψ, χ) = a(ψ, χ), ∀ψ, χ ∈ V_{h},

and the orthogonal projection P_{h} : L_{2}(Ω) → V_{h}, we may write (2.7) as

(2.8)  u_{h,t} + A_{h}u_{h} = P_{h}f(u_{h}), t > 0;  u_{h}(0) = u_{0h}.

The V -ellipticity (2.4) implies that both operators −A and −A_{h} are sectorial on L_{2}(Ω) (uniformly in h), i.e., there are C > 0 and θ ∈ (\frac{1}{2}π, π) such that

(2.9)  ∥(λI + A)^{−1}f∥ + ∥(λI + A_{h})^{−1}P_{h}f∥ ≤ \frac{C}{|λ|},  λ ∈ Σ_{θ}, f ∈ L_{2}(Ω),

where Σ_{θ} denotes the sector {λ ∈ C : λ = ρe^{iφ}, ρ > 0, 0 ≤ |φ| ≤ θ}, see [7, Sections 6–7]. This in turn implies that −A and −A_{h} are the infinitesimal generators of bounded analytic semigroups E(t) = exp(−tA) on L_{2}(Ω) and E_{h}(t) = exp(−tA_{h}) on V_{h}. From the analyticity follows that (D_{t} = ∂/∂t)

(2.10)  ∥D_{t}^{l}E(t)v∥ = ∥A^{l}E(t)v∥ ≤ C_{l}t^{−l}∥v∥,  t > 0, v ∈ L_{2}(Ω), l ≥ 0,

and similarly for E_{h}(t). It also follows that the fractional powers of A and A_{h} are well-defined and we shall use the following equivalences of norms:

∥A^{1/2}v∥ ≈ ∥v∥_{1}, v ∈ D(A^{1/2}) = V; ∥Av∥ ≈ ∥v∥_{2}, v ∈ D(A) = V ∩ H^{2}(Ω),

and the ensuing smoothing property

(2.11) ∥D_{t}^{l}E(t)v∥_{β} ≤ Ct^{−l−(β−α)/2}∥v∥_{α}, t > 0, v ∈ D(A^{α/2}), 0 ≤ α ≤ β ≤ 2, l = 0, 1,

For the discrete operator we have

(2.12)  ∥A_{h}^{1/2}v∥ ≈ ∥v∥_{1},  v ∈ V_{h},

and the smoothing property

(2.13) ∥D_{t}^{l}E_{h}(t)v∥_{β} ≤ Ct^{−l−(β−α)/2}∥v∥_{α}, t > 0, v ∈ V_{h}, 0 ≤ α ≤ β ≤ 1, l = 0, 1.

In order to see that these hold, we write A = A_{1} + A_{2}, where A_{1} is the self-adjoint part of A and we let a_{1}(·, ·) be the corresponding bilinear form, that is, a_{1}(u, v) = ∑_{i,j=1}^{d}(a_{ij}∂u/∂x_{i}, ∂v/∂x_{j}) + (a_{0}u, v). This leads to a corresponding splitting A_{h} = A_{1h} + A_{2h}. It is well-known that

∥A_{1}^{1/2}v∥^{2} = a_{1}(v, v), v ∈ V; ∥A_{1h}^{1/2}v∥^{2} = a_{1}(v, v), v ∈ V_{h},

and that the stated norm equivalences hold for A_{1} and A_{1h}. Since ∥A_{2}v∥ ≤ C∥v∥_{1} for v ∈ V and ∥A_{2h}v∥ ≤ C∥v∥_{1} for v ∈ V_{h}, the general case follows from [11, Theorem 1.4.6].

By Duhamel’s principle we may represent solutions of (2.6) by the integral equation

(2.14)  u(t) = E(t)u_{0} + ∫_{0}^{t} E(t − s)f(u(s)) ds,  t ≥ 0.

Similarly, for solutions of (2.8) we have

(2.15)  u_{h}(t) = E_{h}(t)u_{0h} + ∫_{0}^{t} E_{h}(t − s)P_{h}f(u_{h}(s)) ds,  t ≥ 0.

In order to put equations (2.14) and (2.15) into the framework of [11, Chapter 3] or [17, Section 6.3] we must verify that the operator f : V → L_{2}(Ω) satisfies a local Lipschitz condition. This is the first result of the following lemma. The second result will be used in the error analysis below.

Lemma 2.1. For each bounded set B ⊂ V there is a constant C(B) such that

(2.16)  ∥f(u) − f(v)∥ ≤ C(B)∥u − v∥_{1},  ∀u, v ∈ B,
(2.17)  ∥f(u) − f(v)∥_{−1} ≤ C(B)∥u − v∥,  ∀u, v ∈ B.

Proof. We have in view of (2.3) and the Hölder and Sobolev inequalities

\begin{aligned}∥f(u) − f(v)∥_{L_{2}} & ≤ C(1 + ∥u∥_{L_{q}}^{γ} + ∥v∥_{L_{q}}^{γ})∥u − v∥_{L_{p}} \\ & ≤ C(1 + ∥u∥_{1}^{γ} + ∥v∥_{1}^{γ})∥u − v∥_{1}, \end{aligned}

where \frac{1}{p} + \frac{γ}{q} = \frac{1}{2} with p = q = 6 if d = 3, and with arbitrary p ∈ (1, ∞) if d ≤ 2. This proves (2.16). Moreover,

∥f∥_{−1} = sup_{v∈V} \frac{|(f, v)|}{∥v∥_{1}} ≤ sup_{v∈V} \frac{∥f∥_{L_{r}}∥v∥_{L_{p}}}{∥v∥_{1}} ≤ C∥f∥_{L_{r}},

where \frac{1}{p} + \frac{1}{r} = 1 with p = 6, r = \frac{6}{5} if d = 3, and with arbitrary r ∈ (1, ∞) if d ≤ 2, and

\begin{aligned}∥f(u) − f(v)∥_{L_{r}} & ≤ C(1 + ∥u∥_{L_{q}}^{γ} + ∥v∥_{L_{q}}^{γ})∥u − v∥_{L_{2}} \\ & ≤ C(1 + ∥u∥_{1}^{γ} + ∥v∥_{1}^{γ})∥u − v∥, \end{aligned}

where \frac{1}{2} + \frac{γ}{q} = \frac{1}{r} (thus q = 6 if d = 3). This proves (2.17). □

We may now apply a standard argument to obtain local existence and uniqueness, see [11, Theorem 3.3.3] or [17, Theorem 6.3.1]. The proof is based on applying the contraction mapping principle in the topology of the Banach space C([0, t^{∗}], H^{1}(Ω)) to the integral equations (2.14), (2.15).

Theorem 2.2. For any bounded set B_{0} ⊂ V there is t^{∗} = t^{∗}(B_{0}) such that equation (2.14) has a unique solution u ∈ C([0, t^{∗}], V) for any u_{0} ∈ B_{0}, and equation (2.15) has a unique solution u_{h} ∈ C([0, t^{∗}], V_{h}) for any u_{0h} ∈ B_{0} ∩ V_{h}.

Remark. We may replace the Dirichlet boundary condition in (2.1) by a homogeneous Neumann or Robin boundary condition. The resulting problems can be put into the same abstract framework by a different choice of spaces V, V_{h} and bilinear form a(·, ·), cf. [7]. □

3. Error estimates. We introduce the elliptic projection operator R_{h} : V → V_{h} defined by

(3.1)  a(R_{h}v, χ) = a(v, χ),  ∀χ ∈ V_{h}.

Under the usual regularity assumptions on the triangulation, and in view of the V - ellipticity (2.4) and the elliptic regularity estimate (2.5), we have the error bounds

(3.2)  ∥R_{h}v − v∥ + h∥R_{h}v − v∥_{1} ≤ Ch^{s}∥v∥_{s},  v ∈ V ∩ H^{s}(Ω), s = 1, 2,

see [2]. We may now prove an a priori estimate of the difference between u_{h}(t) and u(t) for t in an arbitrary finite interval. That is, we estimate this difference under the assumption that both u_{h}(t) and u(t) exist and stay bounded in V during a time interval [0, τ].

Theorem 3.1. Let 0 < τ ≤ T and let B ⊂ V be a bounded set. Let u(t) and u_{h}(t) be solutions of (2.14) and (2.15), respectively, such that u(t), u_{h}(t) ∈ B for t ∈ [0, τ]. Then

(3.3)  ∥u_{h}(t) − u(t)∥_{1} ≤ C(B, T)t^{−1/2}(∥u_{0h} − u_{0}∥ + h),  t ∈ (0, τ].

If, in addition, u_{0} belongs to a bounded set B_{1} ⊂ H^{2}(Ω), then

(3.4)  ∥u_{h}(t) − u(t)∥_{1} ≤ C(B, B_{1}, T)(∥u_{0h} − u_{0}∥_{1} + h),  t ∈ [0, τ].

Note that the error bound in (3.3) is of optimal order for t > 0, if u_{0h} is chosen so that ∥u_{0h} − u_{0}∥ ≤ Ch∥u_{0}∥_{1}, for example, if u_{0h} = R_{h}u_{0} or u_{0h} = P_{h}u_{0}. The bound in (3.4) is optimal if ∥u_{0h} − u_{0}∥_{1} ≤ Ch∥u_{0}∥_{2}, for example, if u_{0h} = R_{h}u_{0}.

The situation in (3.4), where u_{0} ∈ V ∩H^{2}(Ω), is referred to as the smooth data case, since the regularity is sufficient for the proof of an optimal order error bound on the interval [0, τ]. The bound (3.3), on the other hand, is an example of a nonsmooth data error estimate, where the initial regularity is insufficient for optimal order convergence at t = 0. Nevertheless we have optimal order convergence for t > 0 due to the smoothing properties of the discrete and continuous solution operators.

In the proof of this we shall need the following regularity estimates.

Theorem 3.2. Let B ⊂ V be a bounded set and 0 < τ ≤ T. If u(t) ∈ B for t ∈ [0, τ], then

(3.5–a)  ∥u(t)∥_{s} ≤ C(B, T)t^{−(s−1)/2},  t ∈ (0, τ], s = 1, 2,
(3.5–b)  ∥u_{t}(t)∥_{s} ≤ C(B, T)t^{−1−(s−1)/2},  t ∈ (0, τ], s = 0, 1.

If, in addition, u_{0} belongs to a bounded set B_{1} ⊂ H^{2}(Ω), then

(3.6–a)  ∥u(t)∥_{2} ≤ C(B, B_{1}, T),  t ∈ [0, τ],
(3.6–b)  ∥u_{t}(t)∥_{s} ≤ C(B, B_{1}, T)t^{−s/2},  t ∈ (0, τ], s = 0, 1.

Proof. In view of (2.16) we have ∥f(u(t))∥ ≤ C(B) for t ∈ [0, τ] (the constant is not necessarily the same as in (2.16)). The proof of (3.5) can now obtained by tracing the constants in [11, Theorem 3.5.2]. The argument is based on a generalization of Gronwall’s lemma: if ϕ(t) satisfies

(3.7)  0 ≤ ϕ(t) ≤ K_{1}t^{−1+α} + K_{2} ∫_{0}^{t}(t − s)^{−1+β}ϕ(s) ds,  t ∈ (0, τ],

then ϕ(t) ≤ C(K_{2}, τ, α, β)K_{1}t^{−1+α} for t ∈ (0, τ], provided that K_{1}, K_{2} ≥ 0, α, β > 0. See [11, Section 7.1] or [5, Lemmas 6.3 and 7.1]; it is clear that the constant C(K_{2}, τ, α, β) is nondecreasing in τ. The case (3.6) of smooth initial data can be treated in a similar way. We omit the details. □

Proof of Theorem 3.1. Following a standard practice we divide the error into two parts:

e(t) ≡ u_{h}(t) − u(t) = (u_{h}(t) − R_{h}u(t)) + (R_{h}u(t) − u(t)) ≡ θ(t) + ρ(t).

In view of (3.2) and (3.5) we have, for j = 0, 1 and s = 1, 2,

(3.8–a) ∥ρ(t)∥_{j} ≤ Ch^{s−j}∥u(t)∥_{s} ≤ C(B, T)h^{s−j}t^{−(s−1)/2},  t ∈ (0, τ],
(3.8–b) ∥ρ_{t}(t)∥ ≤ Ch∥u_{t}(t)∥_{1} ≤ C(B, T)ht^{−1},  t ∈ (0, τ].

It remains to estimate θ(t), which belongs to V_{h}. In view of (2.8), the identity A_{h}R_{h} = P_{h}A and (2.6), we find that

(3.9)  θ_{t} + A_{h}θ = P_{h}(f(u_{h}) − f(u) − ρ_{t}).

Hence

(4.1)  θ(t) = E_{h}(t)θ(0) + ∫_{0}^{t} E_{h}(t − σ)P_{h}(f(u_{h}(σ)) − f(u(σ)) − D_{σ}ρ(σ)) dσ,

where D_{σ} = ∂/∂σ. Integration by parts yields

\begin{aligned}− ∫_{0}^{t/2} E_{h}(t − σ)P_{h}D_{σ}ρ(σ) dσ & = E_{h}(t)P_{h}ρ(0) − E_{h}(t/2)P_{h}ρ(t/2) \\ & + ∫_{0}^{t/2}(D_{σ}E_{h}(t − σ))P_{h}ρ(σ) dσ. \end{aligned}

Hence

\begin{aligned}θ(t) & = E_{h}(t)P_{h}e(0) − E_{h}(t/2)P_{h}ρ(t/2) + ∫_{0}^{t/2}(D_{σ}E_{h}(t − σ))P_{h}ρ(σ) dσ \\ & − ∫_{t/2}^{t} E_{h}(t − σ)P_{h}D_{σ}ρ(σ) dσ + ∫_{0}^{t} E_{h}(t − σ)P_{h}(f(u_{h}(σ)) − f(u(σ))) dσ. \end{aligned}

Using the smoothing property (2.13) of E_{h}(t), the error estimates (3.8) with j = 0, s = 1, and the Lipschitz condition (2.16) we obtain

∥θ(t)∥_{1} ≤ Ct^{−1/2}(∥P_{h}e(0)∥ + ∥ρ(t/2)∥) + C ∫_{0}^{t/2}(t − σ)^{−3/2}∥ρ(σ)∥ dσ
+ C ∫_{t/2}^{t}(t − σ)^{−1/2}∥D_{σ}ρ(σ)∥ dσ

+ C ∫_{0}^{t}(t − σ)^{−1/2}∥f(u_{h}(σ)) − f(u(σ))∥ dσ
≤ C(B, T)t^{−1/2}(∥e(0)∥ + h)
+ C(B, T)h(∫_{0}^{t/2}(t − σ)^{−3/2} dσ + ∫_{t/2}^{t}(t − σ)^{−1/2}σ^{−1} dσ)

+ C(B) ∫_{0}^{t}(t − σ)^{−1/2}∥e(σ)∥_{1} dσ
≤ C(B, T)t^{−1/2}(∥e(0)∥ + h) + C(B) ∫_{0}^{t}(t − σ)^{−1/2}∥e(σ)∥_{1} dσ,

for t ∈ (0, τ]. Together with (3.8) and e = θ + ρ this yields

∥e(t)∥_{1} ≤ C(B, T)t^{−1/2}(∥e(0)∥ + h) + C(B) ∫_{0}^{t}(t − σ)^{−1/2}∥e(σ)∥_{1} dσ, t ∈ (0, τ], and the desired bound follows by the generalized Gronwall lemma, cf. (3.7). The smooth data error bound is proved in a similar way by an estimation based on equation (4.1) without integration by parts. We omit the details. □

Remark. The above argument also gives an error bound in the L_{2} norm:

(3.11)  ∥u_{h}(t) − u(t)∥ ≤ C(B, T)(∥u_{0h} − P_{h}u_{0}∥ + h^{2}t^{−1/2}),  t ∈ (0, τ].

This bound is of optimal order for t > 0, if and only if u_{0h} = P_{h}u_{0}. The proof requires an additional regularity estimate, namely,

∥u_{t}(t)∥_{2} ≤ C(B, T)t^{−3/2}, t ∈ (0, τ],

and one uses the Lipschitz condition (2.17) instead of (2.16) together with the inequality

(3.12)  ∥A_{h}^{−1/2}P_{h}f∥ ≤ C∥f∥_{−1},  f ∈ L_{2}(Ω),

which follows from

∥A_{h}^{−1/2}P_{h}f∥ = sup_{v_{h}∈V_{h}} \frac{|(A_{h}^{−1/2}P_{h}f, v_{h})|}{∥v_{h}∥} = sup_{v_{h}∈V_{h}} \frac{|(f, (A_{h}^{∗})^{−1/2}v_{h})|}{∥v_{h}∥}

= sup_{w_{h}∈V_{h}} \frac{|(f, w_{h})|}{∥(A_{h}^{∗})^{1/2}w_{h}∥} ≤ C sup_{w_{h}∈V_{h}} \frac{|(f, w_{h})|}{∥w_{h}∥_{1}}

≤ C sup_{w∈V} \frac{|(f, w)|}{∥w∥_{1}} = C∥f∥_{−1}.

The interested reader is referred to Section 5, where the details are presented in the context of a fully discrete scheme. □

Remark. A remark about higher order methods is in order. The argument that we used in the proof of the nonsmooth data estimate (3.3) gives optimal order convergence essentially only for piecewise linear finite elements. Optimal error bounds for higher order methods have been proved in [14], [4] by a more complicated technique and in a slightly different setup. However, due to nonlinear effects, there is a limitation on the order of convergence. For example, it was shown in [4] that, in general, ∥u_{h}(t) − u(t)∥ is no smaller than O(h^{4}) for t > 0, if u_{0} ∈ V, no matter what the formal order of the finite element method is. □

4. Long-time behavior. We let S(t) : V → V denote the (local) solution operator of the nonlinear problem (2.14) defined through the relation u(t) = S(t)u_{0}. Similarly, the nonlinear operator S_{h}(t) : V_{h} → V_{h} is defined by u_{h}(t) = S_{h}(t)u_{0h}, where u_{h} is a local solution of equation (2.15).

We first formulate an immediate consequence of the nonsmooth data error estimate of Theorem 3.1. In the terminology of [8] the corollary says that S_{h}(t) conditionally approximates S(t) on B uniformly on compact subsets of R^{+} = (0, ∞).

Corollary 4.1. Let 0 < T_{1} < τ ≤ T_{2} and let B ⊂ V be a bounded set. Then

∥S_{h}(t)v_{h} − S(t)v_{h}∥_{1} ≤ C(B, T_{1}, T_{2})h, t ∈ [T_{1}, τ],

provided that v_{h} ∈ V_{h} is such that S_{h}(t)v_{h}, S(t)v_{h} ∈ B for t ∈ [0, τ].

Before we discuss long-time behavior we must address the question of global existence. For this purpose we assume that solutions of (2.14) are a priori bounded: for any bounded set B_{0} ⊂ V we have ∥u(t)∥_{1} ≤ C(B_{0}) for t ∈ [0, ∞), if u_{0} ∈ B_{0}, or equivalently

(4.1) for any bounded set B_{0} ⊂ V there is a bounded set B_{1} ⊂ V such that S(t)B_{0} ⊂ B_{1} for t ∈ [0, ∞).

By a standard argument this implies that solutions of (2.14) exist for all time. The solution operator S(t) is thus a nonlinear semigroup on V.

It is sometimes possible to prove directly that the solutions of the discrete problem (2.15) are a priori bounded too. But this is not necessary. In fact we shall show by means of the error estimate in Corollary 4.1 that an a priori bound of the form (4.1) for u implies an a priori bound for u_{h} on a finite time interval.

Corollary 4.2. Assume that (4.1) holds and let T > 0 and B_{0} ⊂ V be a bounded set. Then there are h_{0} = h_{0}(B_{0}, T) > 0 and a bounded set B_{2} ⊂ V, depending only on B_{0}, such that for h < h_{0} we have S_{h}(t)(B_{0} ∩ V_{h}) ⊂ B_{2} for t ∈ [0, T].

Proof. The local existence result of Theorem 2.2 implies that there are T_{1} > 0 and a bounded set B ⊂ V such that S_{h}(t)(B_{0} ∩ V_{h}) ⊂ B for t ∈ [0, T_{1}]. Let δ > 0 be arbitrary and let B_{2} be the closed δ-neighborhood of B_{1} ∪ B. Let τ ∈ [0, T] be the largest time such that S_{h}(t)(B_{0} ∩ V_{h}) ⊂ B_{2} for t ∈ [0, τ]. We must show that τ = T. Assume that τ < T. Clearly τ > T_{1}. Moreover, by Corollary 4.1, for any v_{h} ∈ B_{0} ∩ V_{h} we have ∥S_{h}(t)v_{h} − S(t)v_{h}∥_{1} ≤ C(B_{2}, T_{1}, T)h for t ∈ [T_{1}, τ]. Choosing h_{0} so that C(B_{2}, T_{1}, T)h_{0} ≤ δ/2 and recalling that S(t)v_{h} ∈ B_{1} for t ∈ [0, T], we conclude that S_{h}(τ)v_{h} is in the interior of B_{2}. By local existence this contradicts the maximality of τ. □

We may now replace the assumption S(t)v_{h}, S_{h}(t)v_{h} ∈ B in Corollary 4.1 by a bound on the initial value v_{h}.

Corollary 4.3. Assume that (4.1) holds and let 0 < T_{1} < T_{2} and let B_{0} ⊂ V be a bounded set. Then there is h_{0} = h_{0}(B_{0}, T_{2}) > 0 such that for any v_{h} ∈ B_{0} ∩ V_{h} and h < h_{0} we have

∥S_{h}(t)v_{h} − S(t)v_{h}∥_{1} ≤ C(B_{0}, T_{1}, T_{2})h, t ∈ [T_{1}, T_{2}].

Using also the estimate (3.11) we can prove ∥S_{h}(t)v_{h} − S(t)v_{h}∥ ≤ C(B_{0}, T_{1}, T_{2})h^{2}. Similar comments apply to the results below.

In our first application of Corollary 4.3 we follow [12, 13]. Let u be a solution of (2.6). We say that v is a perturbed solution starting at t_{0}, if v(t) = S(t − t_{0})v_{0}, t ≥ t_{0}, with v_{0} near u(t_{0}). The solution u is exponentially stable, if there are numbers δ, T > 0 such that any perturbed solution v(t) = S(t − t_{0})v_{0} with ∥v_{0} − u(t_{0})∥_{1} < δ satisfies

∥v(t) − u(t)∥_{1} ≤ \frac{1}{2}∥v_{0} − u(t_{0})∥_{1}, t ∈ [t_{0} + T, ∞).

Thus T is the time it takes for the perturbation v − u to decay to half its initial size. This would hold, for instance, if u(t) is in the neighborhood of a stable hyperbolic stationary solution. In [12, 13] this property was assumed to hold for certain solutions of the Navier-Stokes equations on the basis of experimental evidence.

Under this assumption we may prove a uniform long-time error estimate.

Theorem 4.4. Assume that (4.1) holds and that u is an exponentially stable solution of (2.6). Assume that u_{0} ∈ V ∩ H^{2}(Ω) and let u_{h} be a solution of (2.8) with ∥u_{0h} − u_{0}∥_{1} ≤ Ch∥u_{0}∥_{2}. Then there are C, h_{0} > 0 such that for h < h_{0} we have

(4.2)  ∥u_{h}(t) − u(t)∥_{1} ≤ Ch,  t ∈ [0, ∞).

Proof. By (4.1) and Corollary 4.2 there is a bounded set B_{0} ⊂ V such that u(t) ∈ B_{0} for t ∈ [0, ∞), and u_{h}(t) ∈ B_{0} for t ∈ [0, T], if h is small. Let B be a δ-neighborhood of B_{0} in V and let C_{1} = C_{1}(B, B_{1}, T) (where u_{0} ∈ B_{1} ⊂ H^{2}(Ω)) be the error constant in the smooth data estimate (3.4) of Theorem 3.1, and C_{2} = C_{2}(B, T, 2T) the error constant in Corollary 4.3. Set C = max(C_{1}, 2C_{2}) and let h_{0} ≤ δ/(2C) be so small that Corollaries 4.2, 4.3 are applicable. Then

(4.5)  ∥u_{h}(t) − u(t)∥_{1} ≤ C_{1}h ≤ Ch,  t ∈ [0, T],
(4.6)  ∥S_{h}(t)v_{h} − S(t)v_{h}∥_{1} ≤ C_{2}h ≤ \frac{1}{2}Ch,  t ∈ [T, 2T],

for v_{h} ∈ B ∩ V_{h}, h < h_{0}.

We shall prove (4.2) by induction. By (4.5) we have the desired bound on the interval [0, T]. To complete the proof it suffices to show that if, for some integer n ≥ 1,

(4.7)  sup_{t∈[0,T_{n}]} ∥u_{h}(t) − u(t)∥_{1} ≤ Ch,  h < h_{0},

where T_{n} = nT, then

(4.8)  sup_{t∈[T_{n},T_{n+1}]} ∥u_{h}(t) − u(t)∥_{1} ≤ Ch,  h < h_{0}.

In order to show this we use the perturbed solution v(t) = S(t − T_{n−1})u_{h}(T_{n−1}) starting at T_{n−1}. We first note that, by our choice of h_{0}, (4.7) implies that

(4.9)  ∥u_{h}(t) − u(t)∥_{1} ≤ δ/2,  t ∈ [0, T_{n}], h < h_{0}.

Hence by exponential stability and (4.7)

\begin{aligned}∥v(t) − u(t)∥_{1} & ≤ \frac{1}{2}∥v(T_{n−1}) − u(T_{n−1})∥_{1} = \frac{1}{2}∥u_{h}(T_{n−1}) − u(T_{n−1})∥_{1} \\ & ≤ \frac{1}{2}Ch, t ∈ [T_{n}, T_{n+1}], h < h_{0}. \end{aligned}

Moreover, (4.9) implies that u_{h}(T_{n−1}) ∈ B and hence by (4.6) we have

\begin{aligned}∥u_{h}(t) − v(t)∥_{1} & ≤ ∥S_{h}(t − T_{n−1})u_{h}(T_{n−1}) − S(t − T_{n−1})u_{h}(T_{n−1})∥_{1} \\ & ≤ \frac{1}{2}Ch, t ∈ [T_{n}, T_{n+1}], h < h_{0}. \end{aligned}

Taken together these estimates show (4.8). □

Remark. All error estimates in [12, 13] are for initial data in V ∩ H^{2}(Ω). Thus nonsmooth data error bounds are used only for higher order methods, but not for piecewise linears. While we compare u_{h}(t) to v(t) = S(t − T_{n−1})v_{0} with v_{0} = u_{h}(T_{n−1}) ∈ V_{h}, the argument in [12, 13] uses a smoother initial value v_{0} = A^{−1}A_{h}u_{h}(T_{n−1}) ∈ V ∩H^{2}(Ω). Instead one needs to prove the a priori bound ∥Au(t)∥ ≤ C, t ∈ [0, ∞). □

In our second application we give a (somewhat simplified) presentation of a result of [8], see also [18, Section I.1.4]. We assume that S(t) has a global attractor A, i.e., A is a compact invariant subset of V, which attracts the bounded sets of V. Thus, for any bounded set B ⊂ V and any ϵ > 0 there is T > 0 such that

S(t)B ⊂ N (A, ϵ), t ∈ [T, ∞),

where N (A, ϵ) denotes the ϵ-neighborhood of A in V. Or equivalently,

δ(S(t)B, A) → 0 as t → ∞,

where δ(A, B) = sup_{a∈A} inf_{b∈B} ∥a − b∥_{1} denotes the unsymmetric semidistance between two subsets A, B of V. For simplicity we assume also that S_{h}(t) has a global attractor A_{h} in V_{h}, although this is actually a consequence of the results of [8].

Theorem 4.5. Under the above assumptions we have δ(A_{h}, A) → 0 as h → 0.

The result means that for any ϵ > 0 there is h_{0} > 0 such that A_{h} ⊂ N (A, ϵ) if h < h_{0}. In the terminology of [8] we say that A_{h} is upper semicontinuous at h = 0. A similar result was proved in [15] by a different method. It is interesting to note that the error bound of Corollary 4.3 is one of the basic assumptions of [15]. The lower semicontinuity was proved in [9] under much stronger assumptions.

Proof. Let ϵ > 0 and B_{0} be a bounded set. We must show that there are T = T (B_{0}, ϵ) > 0 and h_{0} = h_{0}(B_{0}, ϵ) > 0 such that

(4.10)  S_{h}(t)(B_{0} ∩ V_{h}) ⊂ N (A, ϵ),  t ∈ [T, ∞), h < h_{0}.

By (4.1) we have S(t)B_{0} ⊂ B_{1} for t ∈ [0, ∞). Let B = N (B_{1}, ϵ) and choose T such that

(4.11)  S(t)B ⊂ N (A, ϵ/2),  t ∈ [T, ∞).

By Corollary 4.3 we may choose h_{0} such that

(4.12)  ∥S_{h}(t)v_{h} − S(t)v_{h}∥_{1} ≤ C(B, T, 2T)h ≤ ϵ/2,  t ∈ [T, 2T], h < h_{0},

for all v_{h} ∈ B ∩ V_{h}.

We shall prove (4.10) by induction. Take u_{0h} ∈ B_{0} ∩ V_{h} and let u_{h}(t) = S_{h}(t)u_{0h}. (4.11) and (4.12) imply that u_{h}(t) ∈ N (A, ϵ) for t ∈ [T, 2T], h < h_{0}. Assume that, for some integer n ≥ 2,

u_{h}(t) ∈ N (A, ϵ), t ∈ [T, T_{n}], h < h_{0},

where T_{n} = nT. In particular, this implies that u_{h}(T_{n−1}) ∈ B. Let v(t) = S(t − T_{n−1})u_{h}(T_{n−1}). By (4.12) we then have

\begin{aligned}∥u_{h}(t) − v(t)∥_{1} & ≤ ∥S_{h}(t − T_{n−1})u_{h}(T_{n−1}) − S(t − T_{n−1})u_{h}(T_{n−1})∥_{1} \\ & ≤ ϵ/2, t ∈ [T_{n}, T_{n+1}], h < h_{0}. \end{aligned}

Since v(t) ∈ N (A, ϵ/2) for t ∈ [T_{n}, T_{n+1}] by (4.11), we may conclude that u_{h}(t) ∈ N (A, ϵ) for t ∈ [T_{n}, T_{n+1}]. □

5. A completely discrete scheme. In this section we shall show that the above program can be carried through also for a completely discrete scheme based on the backward Euler method. We replace the time derivative in (2.8) by a backward difference quotient ¯∂_{t}U_{n} = (U_{n} − U_{n−1})/k, where k is a time step and U_{n} is the approximation of u_{n} = u(t_{n}) and t_{n} = nk. The discrete solution U_{n} ∈ V_{h} thus satisfies

¯∂_{t}U_{n} + A_{h}U_{n} = P_{h}f(U_{n}), t_{n} > 0; U_{0} = u_{0h}.

Duhamel’s principle yields

(5.1)  U_{n} = E_{kh}^{n}u_{0h} + k ∑_{j=1}^{n} E_{kh}^{n−j−1}P_{h}f(U_{j}),  t_{n} ≥ 0,

where E_{kh} = (I + kA_{h})^{−1}. Since −A_{h} is sectorial uniformly in h, see (2.9), we have the inequality

∥¯∂_{t}^{l}E_{kh}^{n}v∥ = ∥A_{h}^{l}E_{kh}^{n}v∥ ≤ C_{l}t_{n}^{−l}∥v∥, t_{n} > t_{l} ≥ 0, v ∈ V_{h},

which is the discrete analog of (2.10) and can be proved by means of the Dunford-Taylor integral, see [6]. In view of the norm equivalence in (2.12) this leads to a smoothing property analogous to (2.13):

(5.2) ∥¯∂_{t}^{l}E_{kh}^{n}v∥_{β} ≤ Ct_{n}^{−l−(β−α)/2}∥v∥_{α}, t_{n} > t_{l+(β−α)/2}, v ∈ V_{h}, 0 ≤ α ≤ β ≤ 1, l = 0, 1.

In this section we need some regularity estimates in addition to those of Theorem 3.2. In order to prove these we assume that f(x, u) is twice differentiable with respect to u and once with respect to x, and that

(5.3)  \begin{aligned}|f′_{u}(x, u)| + |f′′_{xu}(x, u)| & ≤ C (1 + |u|^{γ}), \\ |f′′_{uu}(x, u)| & ≤ C (1 + |u|^{γ−1}), \end{aligned}  ∀u ∈ R, x ∈ \overline{Ω},

with γ = 2 if d = 3, and γ ∈ [0, ∞) if d = 2. Note that this implies (2.3). In addition to the bounds in Lemma 2.1 we then have the following:

Lemma 5.1. For each bounded set B ⊂ V there is a constant C(B) such that for all u ∈ B ∩ H^{2}(Ω), v ∈ V ∩ H^{2}(Ω) we have f′_{u}(u)v ∈ V with

(5.4)  ∥f′_{u}(u)v∥_{1} ≤ C(B)(∥v∥_{2} + ∥u∥_{2}∥v∥_{1}).

We omit the straightforward proof. This will enable us to obtain the additional regularity estimates that we need in the error analysis.

Theorem 5.2. Let B ⊂ V be a bounded set and 0 < τ ≤ T. If u(t) ∈ B for t ∈ [0, τ], then

(5.5)  ∥u_{tt}(t)∥ + ∥u_{t}(t)∥_{2} ≤ C(B, T)t^{−3/2},  t ∈ (0, τ].

If, in addition, u_{0} belongs to a bounded set B_{1} ⊂ H^{2}(Ω), then

(5.6)  ∥u_{tt}(t)∥ + ∥u_{t}(t)∥_{2} ≤ C(B, B_{1}, T)t^{−1},  t ∈ (0, τ].

If we further assume that Au_{0} − f(u_{0}) belongs to a bounded set B_{2} ⊂ V, then

(5.7)  ∥D_{t}^{l}u(t)∥_{s} ≤ C(B, B_{1}, B_{2}, T)t^{−σ/2},  t ∈ (0, τ], l = 0, 1, 2, s = 1, 2,

where σ = max(2l + s − 3, 0).

Proof. We only sketch the proof of (5.5), which can be based on the fact that v(t) = tu_{t}(t) satisfies

v_{t} + Av = u_{t} + f′_{u}(u)v, t > 0; v(0) = 0,

so that

v(t) = ∫_{0}^{t} E(t − s)(u_{s}(s) + f′_{u}(u(s))v(s)) ds.

Hence, in view of (2.11), (5.4) and (3.5),

\begin{aligned}∥v(t)∥_{2} & ≤ C ∫_{0}^{t}(t − s)^{−3/4}∥u_{s}(s)∥_{1/2} + C ∫_{0}^{t}(t − s)^{−1/2}∥f′_{u}(u(s))v(s)∥_{1} ds \\ & ≤ C(B, T)t^{−1/2} + C(B) ∫_{0}^{t}(t − s)^{−1/2}∥v(s)∥_{2} ds, t ∈ (0, τ]. □ \end{aligned}

We now present our main result. It is evident that the error estimates (5.8), (5.9) can be used in the study of long-time behavior in exactly the same way as in Section 4. The smooth data results are included for completeness.

Theorem 5.3. Let 0 < τ ≤ T and let B ⊂ V be a bounded set. Let u(t) and U_{n} be solutions of (2.14) and (5.1), respectively, such that u(t), U_{n} ∈ B for t, t_{n} ∈ [0, τ]. Then, for k ≤ k_{0}(B), we have

(5.8) ∥U_{n} − u(t_{n})∥ ≤ C(B, T)(∥u_{0h} − P_{h}u_{0}∥ + h^{2}t_{n}^{−1/2} + kt_{n}^{−1/2}), t_{n} ∈ (0, τ],
(5.9) ∥U_{n} − u(t_{n})∥_{1} ≤ C(B, T)(∥u_{0h} − u_{0}∥t_{n}^{−1/2} + ht_{n}^{−1/2} + kt_{n}^{−1}), t_{n} ∈ (0, τ].

If, in addition, u_{0} belongs to a bounded set B_{1} ⊂ H^{2}(Ω), then, for k ≤ k_{0}(B), we have

(5.10) ∥U_{n} − u(t_{n})∥ ≤ C(B, B_{1}, T)(∥u_{0h} − u_{0}∥ + h^{2} + k),  t_{n} ∈ [0, τ],
(5.11) ∥U_{n} − u(t_{n})∥_{1} ≤ C(B, B_{1}, T)(∥u_{0h} − u_{0}∥_{1} + h + kt_{n}^{−1/2}), t_{n} ∈ (0, τ].

If we further assume that Au_{0} − f(u_{0}) belongs to a bounded set B_{2} ⊂ V, then, for k ≤ k_{0}(B), we have

(5.12) ∥U_{n} − u(t_{n})∥_{1} ≤ C(B, B_{1}, B_{2}, T)(∥u_{0h} − u_{0}∥_{1} + h + k),  t_{n} ∈ [0, τ].

Proof. The proof is modeled after the proof of Theorem 3.1. Again we write

e_{n} ≡ U_{n} − u_{n} = (U_{n} − R_{h}u_{n}) + (R_{h}u_{n} − u_{n}) ≡ θ_{n} + ρ_{n},

and the required estimates for ρ_{n} are given in (3.8–a). Moreover, by the error bound (3.2) and the regularity estimates (3.5–b), (5.5),

\begin{aligned}∥¯∂_{t}ρ_{n}∥ & ≤ max_{t_{n−1}≤t≤t_{n}} ∥ρ_{t}(t)∥ ≤ C(B, T)h^{s}t_{n−1}^{−1−(s−1)/2} \\ & ≤ C(B, T)h^{s}t_{n}^{−1−(s−1)/2}, t_{n} ∈ [t_{2}, τ], s = 1, 2, \end{aligned} (5.13)

because \frac{1}{2}t_{n} ≤ t_{n−1} ≤ t_{n} for n ≥ 2, a fact that will be used frequently in the following. For θ_{n} ∈ V_{h} we have similarly to (3.9)

(5.14)  ¯∂_{t}θ_{n} + A_{h}θ_{n} = P_{h}(f(U_{n}) − f(u_{n}) − ¯∂_{t}ρ_{n} + ω_{n}),

where ω_{n} = u_{t}(t_{n}) − ¯∂_{t}u_{n}. Hence

(5.15)  θ_{n} = E_{kh}^{n}θ_{0} + k ∑_{j=1}^{n} E_{kh}^{n−j+1}P_{h}(f(U_{j}) − f(u_{j}) − ¯∂_{t}ρ_{j} + ω_{j}).

We first estimate ∥θ_{n}∥. This estimate will then be employed in the estimation of ∥θ_{n}∥_{1}. Let n_{2} be the integer part of n/2. Summation by parts gives

−k ∑_{j=1}^{n_{2}} E_{kh}^{n−j+1}P_{h} ¯∂_{t}ρ_{j} = E_{kh}^{n}P_{h}ρ_{0} − E_{kh}^{n−n_{2}}P_{h}ρ_{n_{2}} + k ∑_{j=1}^{n_{2}}(¯∂_{t}E_{kh}^{n−j+1})P_{h}ρ_{j},

so that

θ_{n} = E_{kh}^{n}P_{h}e_{0} − E_{kh}^{n−n_{2}}P_{h}ρ_{n_{2}} + k ∑_{j=1}^{n_{2}}(¯∂_{t}E_{kh}^{n−j+1})P_{h}ρ_{j}

− k ∑_{j=n_{2}+1}^{n} E_{kh}^{n−j+1}P_{h} ¯∂_{t}ρ_{j} + k ∑_{j=1}^{n} A_{h}^{1/2}E_{kh}^{n−j+1}A_{h}^{−1/2}P_{h}(f(U_{j}) − f(u_{j}))

+ k ∑_{j=1}^{n_{2}} A_{h}E_{kh}^{n−j+1}(A_{h}^{−1}P_{h} − A^{−1})ω_{j} + k ∑_{j=1}^{n_{2}} A_{h}E_{kh}^{n−j+1}P_{h}A^{−1}ω_{j}

+ k ∑_{j=n_{2}+1}^{n} E_{kh}^{n−j+1}P_{h}ω_{j} ≡ ∑_{i=1}^{8} R_{i}.

The first five terms on the right are handled in a similar way as the corresponding terms in the proof of Theorem 3.1. In this connection we shall use the inequalities

k ∑_{j=1}^{n} t_{n−j+1}^{−1+α}t_{j}^{−1+β} ≤ C_{α,β}t_{n}^{−1+α+β},  α, β > 0,

k ∑_{j=1}^{n_{2}} t_{n−j+1}^{−α}t_{j}^{−1+β} ≤ t_{n−n_{2}+1}^{−α}k ∑_{j=1}^{n_{2}} t_{j}^{−1+β} ≤ C_{α,β}t_{n}^{−α+β},  α, β > 0,

which follow by comparison with the integral ∫_{0}^{t}(t − s)^{−1+α}s^{−1+β} ds.

Thus, using the smoothing property (5.2), the error bounds (3.8–a), (5.13), and also (3.12), (2.17) in the fifth term, we have for s = 1, 2

∑_{i=1}^{5} ∥R_{i}∥ ≤ C(∥P_{h}e_{0}∥ + ∥ρ_{n_{2}}∥) + Ck ∑_{j=1}^{n_{2}} t_{n−j+1}^{−1}∥ρ_{j}∥

+ Ck ∑_{j=n_{2}+1}^{n} ∥¯∂_{t}ρ_{j}∥ + Ck ∑_{j=1}^{n} t_{n−j+1}^{−1/2}∥f(U_{j}) − f(u_{j})∥_{−1}

≤ C∥P_{h}e_{0}∥ + C(B, T)h^{s}t_{n}^{−(s−1)/2}

+ C(B, T)h^{s} (k ∑_{j=1}^{n_{2}} t_{n−j+1}^{−1}t_{j}^{−(s−1)/2} + k ∑_{j=n_{2}+1}^{n} t_{j}^{−1−(s−1)/2})

+ C(B)k ∑_{j=1}^{n} t_{n−j+1}^{−1/2}∥e_{j}∥

≤ C∥P_{h}e_{0}∥ + C(B, T)h^{s}t_{n}^{−(s−1)/2} + C(B)k ∑_{j=1}^{n} t_{n−j+1}^{−1/2}∥e_{j}∥, t_{n} ∈ (0, τ].

(Note that the above argument works also when n = 1, n_{2} = 0.) For the terms involving ω_{j} we need the following inequalities:

(5.16–a)  ∥A^{−l}ω_{j}∥_{s} ≤ C(B, T)t_{j}^{l−1−(s−1)/2},  t_{j} ∈ (0, τ], l = 0, 1,
(5.16–b)  ∥A^{−l}ω_{j}∥ ≤ C(B, T)kt_{j}^{l−3/2},  t_{j} ∈ (0, τ], l = 0, 1,

where we allow s = 0 if l = 0, and s = 1, 2 if l = 1. In fact, using regularity estimates from Theorems 3.2, 5.2,

∥A^{−l}ω_{j}∥_{s} ≤ max_{t_{j−1}≤t≤t_{j}} ∥A^{−l}u_{t}(t)∥_{s} ≤ C(B, T)t_{j−1}^{l−1−(s−1)/2} ≤ C(B, T)t_{j}^{l−1−(s−1)/2},

for t_{j} ∈ [t_{2}, τ], since, by equation (2.6) and (2.16),

∥A^{−1}u_{t}∥_{s} ≤ ∥u∥_{s} + ∥A^{−1}f(u)∥_{s} ≤ ∥u∥_{s} + C∥f(u)∥ ≤ ∥u∥_{s} + C(B).

Similarly,

\begin{aligned}∥A^{−l}ω_{j}∥ & = ∥k^{−1} ∫_{t_{j−1}}^{t_{j}}(t − t_{j−1})A^{−l}u_{tt}(t) dt∥ ≤ k max_{t_{j−1}≤t≤t_{j}} ∥A^{−l}u_{tt}(t)∥ \\ & ≤ C(B, T)kt_{j−1}^{l−3/2} ≤ C(B, T)kt_{j}^{l−3/2}, \end{aligned}

for t_{j} ∈ [t_{2}, τ], since

∥A^{−1}u_{tt}∥ ≤ ∥u_{t}∥ + ∥A^{−1}f′_{u}(u)u_{t}∥ ≤ ∥u_{t}∥ + C∥f′_{u}(u)u_{t}∥_{−1} ≤ C(B)∥u_{t}∥,

cf. (3.12) and (2.17). Moreover, under the stated restrictions on s,

\begin{aligned}∥A^{−l}ω_{1}∥_{s} & ≤ k^{−1} ∫_{0}^{k} t∥A^{−l}u_{tt}(t)∥_{s} dt ≤ C(B, T)k^{−1} ∫_{0}^{k} t^{l−1−(s−1)/2} dt \\ & = C(B, T)t_{1}^{l−1−(s−1)/2} = C(B, T)kt_{1}^{l−2−(s−1)/2}, \end{aligned}

and we conclude that the above estimates of ω_{j} hold also for j = 1.

Hence, since A_{h}^{−1}P_{h} −A^{−1} = (R_{h} −I)A^{−1}, we have by (3.2) and (5.16), for s = 1, 2,

∥R_{6}∥ ≤ Ck ∑_{j=1}^{n_{2}} t_{n−j+1}^{−1}∥(R_{h} − I)A^{−1}ω_{j}∥ ≤ Ch^{s}k ∑_{j=1}^{n_{2}} t_{n−j+1}^{−1}∥A^{−1}ω_{j}∥_{s}

≤ C(B, T)h^{s}k ∑_{j=1}^{n_{2}} t_{n−j+1}^{−1}t_{j}^{−(s−1)/2} ≤ C(B, T)h^{s}t_{n}^{−(s−1)/2},

∥R_{7}∥ ≤ Ck ∑_{j=1}^{n_{2}} t_{n−j+1}^{−1}∥A^{−1}ω_{j}∥ ≤ C(B, T)k^{2} ∑_{j=1}^{n_{2}} t_{n−j+1}^{−1}t_{j}^{−1/2} ≤ C(B, T)kt_{n}^{−1/2},

and

∥R_{8}∥ ≤ Ck ∑_{j=n_{2}+1}^{n} ∥ω_{j}∥ ≤ C(B, T)k^{2} ∑_{j=n_{2}+1}^{n} t_{j}^{−3/2} ≤ C(B, T)kt_{n}^{−1/2}.

Taken together these estimates show

∥e_{n}∥ ≤ C(B, T)(∥P_{h}e_{0}∥ + h^{s}t_{n}^{−(s−1)/2} + kt_{n}^{−1/2}) + C(B)k ∑_{j=1}^{n} t_{n−j+1}^{−1/2}∥e_{j}∥, t_{n} ∈ (0, τ],

and the generalized discrete Gronwall lemma [5, Lemma 7.1] yields, for k ≤ k_{0}(B),

(5.17) ∥e_{n}∥ ≤ C(B, T)(∥P_{h}e_{0}∥ + h^{s}t_{n}^{−(s−1)/2} + kt_{n}^{−1/2}),  t_{n} ∈ (0, τ], s = 1, 2,

and, since P_{h}e_{0} = u_{0h} − P_{h}u_{0}, this proves (5.8).

We now turn to the estimation of ∥θ_{n}∥_{1}. Because of the singularity t_{n}^{−1} in the final result we multiply equation (5.14) by t_{n}:

¯∂_{t}(t_{n}θ_{n}) + A_{h}(t_{n}θ_{n}) = θ_{n−1} + t_{n}P_{h}(f(U_{n}) − f(u_{n}) − ¯∂_{t}ρ_{n} + ω_{n}).

Hence

t_{n}θ_{n} = k ∑_{j=1}^{n} E_{kh}^{n−j+1} (θ_{j−1} + t_{j}P_{h}(f(U_{j}) − f(u_{j}) − ¯∂_{t}ρ_{j} + ω_{j})) ,

so that

t_{n}∥θ_{n}∥_{1} ≤ Ck ∑_{j=1}^{n} t_{n−j+1}^{−1/2}(∥θ_{j−1}∥ + t_{j}∥f(U_{j}) − f(u_{j})∥ + t_{j}∥¯∂_{t}ρ_{j}∥ + t_{j}∥ω_{j}∥).

Here we have

∥θ_{j−1}∥ ≤ C(B, T)(∥e_{0}∥ + h + kt_{j}^{−1/2}), t_{j} ∈ (0, τ].

For t_{j} ∈ [t_{2}, τ] this follows from (5.17) (recall that \frac{1}{2}t_{j} ≤ t_{j−1}) and for j = 1 we have ∥θ_{0}∥ ≤ ∥e_{0}∥ + ∥ρ_{0}∥ ≤ ∥e_{0}∥ + C(B)h. Moreover,

t_{j}∥¯∂_{t}ρ_{j}∥ ≤ C(B, T)h, t_{j} ∈ (0, τ].

For t_{j} ∈ [t_{2}, τ] this follows from (5.13) and for j = 1 we have simply t_{1}∥¯∂_{t}ρ_{1}∥ ≤ ∥ρ_{1}∥ + ∥ρ_{0}∥ ≤ C(B)h. Hence, using also (5.16–b) we obtain

\begin{aligned}t_{n}∥e_{n}∥_{1} & ≤ C(B, T)k ∑_{j=1}^{n} t_{n−j+1}^{−1/2}(∥e_{0}∥ + h + kt_{j}^{−1/2}) + C(B)k ∑_{j=1}^{n} t_{n−j+1}^{−1/2}(t_{j}∥e_{j}∥_{1}) \\ & ≤ C(B, T)(∥e_{0}∥t_{n}^{1/2} + ht_{n}^{1/2} + k) + C(B)k ∑_{j=1}^{n} t_{n−j+1}^{−1/2}(t_{j}∥e_{j}∥_{1}), t_{n} ∈ (0, τ], \end{aligned}

and Gronwall’s lemma yields, for k ≤ k_{0}(B),

t_{n}∥e_{n}∥_{1} ≤ C(B, T)(∥e_{0}∥t_{n}^{1/2} + ht_{n}^{1/2} + k), t_{n} ∈ (0, τ],

which is (5.9). The smooth data error bounds can be proved in a similar way using the above ingredients. We omit the details. □

Remark. The above argument works also for the semi-implicit scheme

¯∂_{t}U_{n} + A_{h}U_{n} = P_{h}f(U_{n−1}), t_{n} > 0; U_{0} = u_{0h}.

In this case equation (5.14) is replaced by

¯∂_{t}θ_{n} + A_{h}θ_{n} = P_{h}(f(U_{n−1}) − f(u_{n}) − ¯∂_{t}ρ_{n} + ω_{n}),

and we make the estimation (for l = 0, 1)

\begin{aligned}∥k ∑_{j=1}^{n} E_{kh}^{n−j+1}P_{h}(f(U_{j−1}) − f(u_{j}))∥_{l} & ≤ Ck ∑_{j=1}^{n} t_{n−j+1}^{−1/2}∥U_{j−1} − u_{j}∥_{l} \\ & ≤ Ck ∑_{j=1}^{n} t_{n−j+1}^{−1/2}(∥e_{j−1}∥_{l} + k∥¯∂_{t}u_{j}∥_{l}), \end{aligned}

where

k∥¯∂_{t}u_{j}∥_{l} ≤ C(B, T)t_{j}^{−1−(l−1)/2}, t ∈ (0, τ].

Note that the condition k ≤ k_{0}(B) is now not necessary in the Gronwall argument. □

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Department of Mathematics, Chalmers University of Technology, S–412 96 Göteborg, Sweden

E-mail address: stig@math.chalmers.se